
where the definition of the second covariant derivative \(\nabla ^2\bar{f}\left(\bar{{\cal X}}\right)\left[\cdot ,\cdot \right]\)  can be found for example in [1]}.
Approximating \( \nabla ^2\bar{f}\left(\bar{{\cal X}}\right)\left[\bar{\xi }_{\bar{{\cal X}}},{\cal P}_{\bar{{\cal X}}}^{{\cal H}} \bar{\eta }\right]\)  by \(\left\langle \mathrm {D} \bar{F}\left(\bar{{\cal X}}\right)\left[\bar{\xi }_{\bar{{\cal X}}}\right], \mathrm {D} \bar{F}\left(\bar{{\cal X}}\right)\left[{\cal P}_{\bar{{\cal X}}}^{{\cal H}} \bar{\eta } \right]\right\rangle \)  yields the Gauss-Newton equation [1]}
\(0 &= \left\langle \mathrm {D} \bar{F}\left(\bar{{\cal X}}\right)\left[\bar{\eta }\right],\bar{F}\left(\bar{{\cal X}}\right)\right\rangle + \left\langle \mathrm {D} \bar{F}\left(\bar{{\cal X}}\right)\left[\bar{\eta } \right], \mathrm {D} \bar{F}\left(\bar{{\cal X}}\right)\left[\bar{\xi }_{\bar{{\cal X}}}\right]\right\rangle , ~\text{for all}~ \bar{\eta }\in {\cal H}_{\bar{{\cal X}}}.\) 
